Arc Length of Curve r = sin^2(theta/2)

[flakine]

Find Arc Length of Curve r=sin^2(θ/2) from θ=0 to θ=pi/2

√((sin^2(θ/2))^2 + 2(2sin(θ/2))(1/2))

I'm lost on how to pull this one together!

 

[galactus]

Using the arc length formula for polar coordinates.

\(\displaystyle \L\\\int{\sqrt{r^{2}+(\frac{dr}{d{\theta}})^{2}}}d{\theta}\)

\(\displaystyle \frac{dr}{d{\theta}}=sin(\frac{\theta}{2})cos(\frac{\theta}{2})\)

\(\displaystyle \L\\\int_{0}^{\frac{\pi}{2}}{\sqrt{(sin^{2}(\frac{\theta}{2}))^{2}+sin^{2}(\frac{\theta}{2})cos^{2}(\frac{\theta}{2})}}d{\theta}\)

This looks monstrous, but it simplifies nicely.

\(\displaystyle \L\\\int_{0}^{\frac{\pi}{2}}{\sqrt{(sin^{2}(\frac{\theta}{2}))(sin^{2}(\frac{\theta}{2})+cos^{2}(\frac{\theta}{2}))}}d{\theta}\)

\(\displaystyle \L\\\int_{0}^{\frac{\pi}{2}}sin(\frac{\theta}{2})d{\theta}\)